JSCE Journal of Earthquake Engineering
SEISMIC EVALUATION OF GROUP PILE
FOUNDATION WITH M1 AND M2 MODEL
Feng ZHANG1, Atsushi YASHIMA2, Kazuhide SAWADA3 and Yoshitaka NODA4
1
Associate Professor, River Basin Research Center, Gifu University, Gifu, 501-1193, Japan.
zhang-f@cive.gifu-u.ac.jp
2
Professor, Dept. of Civil Engineering, Gifu University, Gifu, 501-1193, Japan. yashima@cc.gifu-u.ac.jp
3
Instructor, Dept. of Civil Engineering, Gifu University, Gifu, 501-1193, Japan. sawada@cc.gifu-u.ac.jp
4
Former student, Dept. of Civil Engineering, Gifu University, Gifu, 501-1193, Japan
In this paper, a series of numerical analyses using dynamic finite element method on simplified model
and full model are conducted to investigate the seismic behaviors of group-pile foundation. In the
analyses, a beam theory proposed for RC material, in which the axial-force dependency in the nonlinear
moment-curvature relation can be considered properly, is used. Two kinds of models for a ground-pile
foundation-superstructure system are used. One is called as simplified model, in which, the interaction
between piles and ground is represented by springs and the ground-pile foundation-superstructure system
is simplified to a frame-spring model (M1 model). Another model is call as full model, in which, the
system is modeled with three-dimensional finite elements without simplification (M2 model). The
purpose of the research is to verify the applicability of the dynamic analysis with M1 model that can be
commonly used in daily seismic design without much difficult. The analyses with M1 and M2 models on
a highway bridge with a 9-pile foundation are conducted to check the accuracy of the analysis with
simplified model.
Key words: Group pile, Foundation, Dynamic, FEM, Plasticity
with M2 model is still a little bit of difficulty for
engineers. The commonly used model is usually a
frame-spring model (M1 model). Many simplified
assumptions, however, have been included in the
M1 model and therefore it is necessary to verify the
accuracy of the analysis with M2 model.
In a common analysis, a very important fact, that is,
the influence of axial force on the stiffness and the
bending strength of RC piles that greatly affects the
nonlinearity of piles, is neglected. The reason why
the influence of axial force was not considered is
that it is difficult to model the influence under cyclic
loading condition within the framework of common
beam theory. For this reason, by introducing a new
weak form of the equilibrium equation for beams
(Zhang & Kimura, 2002), the interaction between
the bending moment and the axial force can be
properly evaluated under generalized loading
conditions.
As to the nonlinearity of soil, a kinematic hardening
elastoplastic constitutive model using the concept of
1. INTRODUCTION
It is known that during a strong earthquake, the
dynamic behavior of a group-pile foundation is not
only related to the inertial force come from the
superstructures but also to the deformation of the
surrounding ground. During the Hyogoken-Nambu
earthquake, it is found from field observations
(Horikoshi et al., 1996) that even in the absence of a
superstructure, piles failed because of the
deformation of the surrounding ground. The
moments developed due to the deformation of
ground are usually referred to as kinematic
moments.
On the other hand, dealing with a full system, which
consists of superstructures, a foundation and a
ground, in a numerical dynamic analysis in time
domain is usually thought to be effective and
executable nowadays when the nonlinearity of the
superstructure, the piles and the soils is considered.
In daily seismic design, however, dynamic analysis
1
subloading, known as tij subloading model
(Chowdhury et al., 1999), is adopted for soils in the
3-D dynamic analysis conducted in this paper.
One of the purposes of the paper is to provide an
accurate numerical method of evaluating the
dynamic behavior of a group-pile foundation based
on 3-D finite element analysis for a superstructure
-foundation-ground system, in which the
nonlinearity of a pile is described by the axial-force
dependent model and soils are described with the tij
subloading model. Another purpose of the paper is
to check the accuracy of the analysis with M1 model,
based on the condition that the dynamic analysis on
M2 model is an accurate method in seismic design
of group-pile foundation. In the paper, all the
numerical analyses are conducted in the same
program named DGPILE-3D (Zhang et al., 2000).
geometrical and loading condition.
In M1 model, three piles in each row are combined
to one so that the simplified model can be used in
two-dimensional analysis. The springs representing
the interaction between pile and surrounding ground
in horizontal and vertical directions, the interaction
between footing and the surrounding ground, and
the shear spring representing the connection of free
ground slices, are evaluated according to the Design
Codes of Foundations and Earth-Retaining
Structures of Japan Railway (Japan Ministry of
Transportation, 1997). Detailed description about
the evaluation of these equivalent springs can be
referred to corresponding references, e.g., Mori,
1997 and Noda, 2003. Figure 1 shows the
simplified M1 model.
In M2 model, due to the above-mentioned
symmetrical conditions, only six piles are
considered as shown in Figure 2. In order to make
the comparison between the analyses using M1 and
M2 models more easily, the first row of piles are
call as Pile1, and so on. The boundary condition of
M2 model is that: (a) the bottom of the ground is
fixed; (b) the vertical boundaries parallel to the
XOZ plane are roll boundaries; (c) an
equal-displacement-boundary condition is used
between the two other side boundaries The
boundary condition of the piles is that the head of
the pile is fixed with the footing and the toe of the
pile is free.
2. SEISMIC RESPONSES OF GROUND-PILE
FOUNDATION USING M1 AND M2
MODELS
In order to avoid meaningless comparison, the
calculations with M1 and M2 models are conducted
with the same program name as DGPILE-3D.
In the calculation, an elevated highway bridge with
a group-pile foundation made of 3×3 cast-in-place
reinforced concrete piles, in which the
center-to-center distance of the group piles is 2.5D,
is considered. Table 1 shows the physical and
geometrical properties of the piles. The ground is
composed of six layers of sandy soils and clayed
soils. Table 2 lists the material parameters of the
soils.
Mass of superstructure
Transverse direction
of bridge
Beam element
Kf
Mass of
free ground
Table 1 Physical and geometrical properties of pile
Young’s
Young’s
Sectional
modulus
modulus
moment
(Reinforcement) (Concrete)
L (m) σｙ(MPa)
EC (kPa)
I (m4)
ES (kPa)
15.0
295
2.000E+08 2.500E+07 1.018E-01
Diameter Length
D (m)
1.2
Table 2
Layer
AS1
AC1
AS2
AC2
AS3
DS
Kho
Yield
strength
Kg
Kv1
Khi
Material parameters of soils
Thickness Density Poisson’s
(m)
(g/cm3) Ratio
H
ρ
ν
2.2
1.7
0.30
2.6
1.7
0.40
4.0
1.7
0.30
3.5
1.7
0.40
3.5
1.9
0.30
1.4
1.9
0.30
Void Stress ratio Compression Swelling
ratio at failure
index
index
e0
Rf
Ct
Ce
0.93
4.5
0.0234
0.0140
0.88
3.5
0.0191
0.0124
0.93
4.6
0.0124
0.0092
0.88
3.5
0.0168
0.0097
0.87
4.7
0.0084
0.0060
0.65
-------
Kv2
Kg : Shear spring of free ground
Kv1: Vertical frictional spring between pile and
ground
Kv2: Vertical spring of pile tip
Kf: Spring between footing and ground
Khi: Horizontal interaction spring between pile and
inner ground
Kho: Horizontal interaction spring between pile and
outer ground
For simplicity, earthquake wave is input in one
direction so that the domain can be reduced to half
volume because of the symmetric conditions of both
Figure 1
Simplified model of ground-pile foundationsuperstructure system (M1 model)
2
4
Acceleration(m/s2)
2.1 Comparison of seismic responses of free
ground using M1 and M2 models under elastic
condition
Before conducting the comparison between the
analyses of group-pile foundation, the seismic
responses of the free ground with M1 and M2
models under elastic condition is investigated. In
M1 model, the free ground is simply modeled with a
column of masses and springs as shown in the right
side of Figure 1. In M2 model, the ground is
modeled with a three-dimensional finite element
mesh without piles and column as shown in Figure
2. Figure 3 shows the input wave of earthquake.
A direct integration method of Newmark-β is
adopted in the dynamic analysis. A Rayleigh type of
damping is adopted and the damping factors of the
structures and the ground are assumed as 2%, in the
dynamic analysis of a full system. Although the
stiffness of the ground, the piles, and the column
may change because of the nonlinearity of these
materials, the viscous matrix calculated from the
Rayleigh type of damping is assumed to be constant
irrespective of the changes in the stiffness matrix. In
calculating the viscous matrix, an eigenvalue
analysis for the full system is conducted to evaluate
the first two eigenvalues. The eigenvalue analysis is
conducted with a hybrid of Jacobian and subspace
methods. In the dynamic analysis, the time interval
of the integration is 0.01 sec. In the calculation, 20
seconds of the main vibration of the input wave is
calculated.
Figure 4 shows the comparisons of response
acceleration and displacement at the surface of the
free ground. It can be seen from the figure that
under elastic condition, the results from M1 and M2
models are totally the same, implying that the
simplification involved in M1 model is completely
acceptable in seismic evaluation of ground under
elastic condition.
Z
Y
Figure 3
20
30
Acceleration(m/s )
2
15
10
5
0
-5
-10
-20
M1
M2
0
5
10
Time(sec)
15
10
Time(sec)
15
20
(a) Acceleration at the surface of ground
Displacement(m)
0.4
0.3
0.2
0.1
0
-0.1
-0.2
M1
M2
-0.3
-0.4
0
5
20
(b) Displacement at the surface of ground
Figure 4 Comparison of seismic responses of free ground
using M1 and M2 models under elastic condition
2.2 Comparison of seismic responses of full
system using M1 and M2 models
In the dynamic analyses of the ground-pile
foundation-superstructure system using M1 and M2
models, the nonlinearity of the ground, the piles and
the superstructure is fully considered. In M1 model,
all the equivalent springs are modeled with tri-linear
model considering the hysteresis of loading and
unloading. The nonlinearity of pile is simulated by
two kinds of models; one is tri-linear model that
cannot take into consideration the axial-force
dependency (Briefly, M1-Tri), another is AFD
model (Briefly M1-AFD). In M2 model, the soils
are simulated with tij subloading clay model and
original tij sand model. The pile is simulated with
AFD model. The superstructure, or column, is
simulated by tri-linear model in both M1 and M2
models.
Figure 5 shows the comparison of response
accelerations at the top of column and the bottom of
column. It is found that the maximum accelerations
at the top of column from M1 and M2 models are in
the same order. While at the bottom, the response
Unit: ｍ
Arrangement of piles
Pile2
Time (sec)
Input wave
-15
Node：10106
Element：8704
Pile1
Figure 2
10
20
X
O
-2
0
17.2
85.0
0
-4
Vibration direction
42.5
2
Pile3
Finite element mesh (M2 model)
3
4
30
M 1 ( AF D)
M2
20
2
1
Moment(MN*m)
2
Acceleration(m/s )
3
0
-1
-2
-3
10
0
-1 0
-2 0
-4
0
2
4
6
T im e ( se c )
8
1 0
-3 0
M 1( tr i)
M 1( AF D)
(a) Top of pier
-4 0
0
4
2
Acceleration(m/s )
2
M 1( AF D)
M 2
3
4
6
T im e (s e c )
10
Figure 8 Comparison of time history of bending moment at
the bottom of column using M1-AFD and M1-Tri models
2
1
0
-1
-2
Moment(MN*m)
-3
-4
0
2
4
6
T im e ( se c )
8
1 0
(b) Bottom of pier
Figure 5 Comparison of response accelerations using M1 -AFD
and M2 models
G.L.-2.700m
5
4
3
2
1
0
-1
-2
-3
-4
-5
Pile1
Pile2
Pile3
0
0 .5
2
4
6
8
10
Time(sec)
0 .4
(a) Bending moment
0 .3
G.L.-2.700m
0 .2
0 .1
8
0
Axial force(MN)
Displacement(m)
8
- 0 .1
- 0 .2
- 0 .3
DI SP -A FD
DI SP -M 2
- 0 .4
- 0 .5
0
2
4
6
T im e ( se c )
8
1 0
(a) Top of pier
Pile1
Pile2
Pile3
6
4
2
0
-2
-4
0 .5
0
2
4
0 .4
6
0 .2
0 .1
8
10
G.L.-2.700m
3
0
Shearing force(MN)
Displacement(m)
Time(sec)
(b) Axial force
0 .3
- 0 .1
- 0 .2
- 0 .3
M 1( AF D)
M 2
- 0 .4
- 0 .5
0
2
4
6
T im e ( se c )
8
1 0
(b) Bottom of pier
Pile1
Pile2
Pile3
2
1
0
-1
-2
-3
Figure 6 Comparison of response displacements using M1
-AFD and M2 models
0
(c) Shear force
2
4
Time(sec)
6
8
10
Figure 9 Time histories of sectional forces at pile head using
M2 model
30
20
Moment(MN*m)
10
accelerations are quite different. The same tendency
can be observed in the response displacements from
M1 and M2 models, as shown in Figure 6. The
displacement evaluated with M1 model is much
larger than that of M2 model while the acceleration
evaluated with M1 model is much smaller than that
of M2 model. This is thought to be the reason that in
M1 model, the stiffness of ground will decrease
dramatically when the stress overpass the yielding
point due to the adoption of tri-linear model. As the
results of the difference between the response
accelerations and displacements, the bending
moments and the moment-curvature relations at the
bottom of column, are also quite different, as shown
in Figure 7.
Figure 8 shows the comparison of time history of
bending moment at the bottom of column from M1-
0
-1 0
-2 0
-3 0
M1( AF D)
M2
-4 0
0
2
4
6
T im e ( se c )
8
1 0
(a) Time history of bending moment at the bottom of column
3 0
Moment(MN*m)
2 0
1 0
0
-1 0
-2 0
-3 0
-4 0
- 0 .0 0 8
M1(AFD)
M2
- 0 .0 0 6
- 0 .0 0 4
- 0 .0 0 2
0
C u r v a tu r e ( 1 / m )
0 .0 0 2
0 .0 0 4
(b) M-φ relation at the bottom of column
Figure 7 Comparison of moment-curvature relation using
M1-AFD and M2 models
4
Moment(MN*m)
G.L.-2.700m
5
4
3
2
1
0
-1
-2
-3
-4
-5
AFD and M1-Tri models. It is found that the
difference of the nonlinear model used for piles in
M1 model has little influence on the responding
moment at the bottom of column (superstructure),
though it may greatly affect the response of
sectional forces of piles as can be seen in Figures
9-11.
Figures 9-11 show the time histories of sectional
forces at pile head using different models. In the
analysis with M2 model, the difference of sectional
forces between different piles is clearly described.
In M1-AFD model, due to the adoption of AFD
model for piles, the difference of sectional forces
between different piles can also be described to
some extent. In M1-Tri model, however, the
difference cannot be described at all. The above
results indicate that it is very important to introduce
a proper model for piles both in M1 and M2 models.
Pile1
Pile2
Pile3
0
2
4
Time(sec)
6
8
10
(a) Bending moment
G.L.-2.700m
Axial force(MN)
8
Pile1
Pile2
Pile3
6
4
2
0
-2
-4
0
2
4
Time(sec)
6
8
10
(b) Axial force
G.L.-2.700m
Shearing force(MN)
3
Pile1
Pile2
Pile3
2
1
M1-tri(Pile1)t=3.82s
M1-tri(Pile2)t=3.82s
M1-tri(Pile3)t=3.82s
M1-AFD(Pile1)t=3.82s
M1-AFD(Pile2)t=3.82s
M1-AFD(Pile3)t=3.82s
M2(Pile1)t=3.34s
M2(Pile2)t=3.34s
M2(Pile3)t=3.34s
My
Mu
0
-1
-2
-3
0
2
4
(c) Shear force
Time(sec)
6
8
10
2
G.L.-2.700m
-5
-4
-3
-2
-1
0
1
2
3
4
5
Pile1
Pile2
Pile3
4
6
0
2
4
Time(sec)
6
8
Depth(m)
Moment(MN*m)
Figure 10 Time histories of sectional forces at pile head using
M1 (AFD) model
10
(a) Bending moment
8
10
12
G.L.-2.700m
Axial force(MN)
8
14
Pile1
Pile2
Pile3
6
16
4
2
18
0
-6
-2
0
2
4
6
Moment(MN*m)
Figure 12 Comparison of the distributions in the maximum
bending moment of piles using M1 and M2 models
-2
-4
0
2
4
(b) Axial force
Time(sec)
6
10
G.L.-2.700m
3
Shearing force(MN)
8
-4
Figure 12 shows the comparison of the distributions
in the maximum bending moment of piles obtained
from different analyses with M1 and M2 models. It
is found that the bending moment from M2 model
is, on the whole, much larger than that from M1
model. It is also known that the distribution of the
bending moment from M2 model changed its
direction along the depth, which is thought to be
usual in group-pile foundations. The results from
M1 models, however, did not change along the
depth. This is thought to be the reason that the
Pile1
Pile2
Pile3
2
1
0
-1
-2
-3
0
(c) Shear force
2
4
Time(sec)
6
8
10
Figure 11 Time histories of sectional forces at pile head using
M1 (Tri) model
5
interactions between the piles and ground are
underestimated in the analyses with M1 models. For
this reason, it should be pointed out that the seismic
evaluation using M1 model has the risk of
underestimation of sectional forces due to
earthquake vibration.
REFERENCES
1)
2)
3. CONCLUSIONS
3)
Under elastic condition, the seismic behavior of free
ground from M1 and M2 models are totally the
same. Therefore, the simplification involved in M1
model is completely acceptable in seismic
evaluation of free ground under elastic condition.
Under elasto-plastic condition, however, there is big
difference between the analyses using M1 and M2
models.
The displacements at the top and bottom of column
evaluated with M1 model are much larger than that
of M2 model while the accelerations evaluated with
M1 model is much smaller than that of M2 model.
The bending moment from M2 model is, on the
whole, much larger than that from M1 model. The
distribution of the bending moment from M2 model
changed its direction along the depth, while the
results from M1 models did not change along the
depth. These differences are found to be caused by
the difference of the deformation of the ground
evaluated by different models. The seismic
evaluation using M1 model has the risk of
underestimation of sectional forces due to
earthquake vibration. Therefore, further research
should be made to improve the modeling of M1
model in its application.
4)
5)
6)
7)
8)
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Nakai, T., Matsuoka, H., Okuno, N. and Tsuzuki, K.
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Noda, Y. 2003, Seismic evaluation of group-pile
foundation using M1, M2 models, Bachelor Thesis, Gifu
University, Japan (in Japanese)
Zhang, F., Kimura, M., Nakai, T. and Hoshikawa, T.
2000. Mechanical behavior of pile foundations subjected
to cyclic lateral loading up to the ultimate state, Soils and
Foundations, Vol. 40, No.5, 1-18.
Zhang, F. and Kimura, M. 2002. Numerical Prediction of
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(Received June 28, 2003)
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